Remark 3.2 Note that the above scheme is implicit in the diffusion term. This is done to ensure a simple theoretical analysis for merely Hölder continuous \(\varphi \) . If we choose an explicit scheme, we need to rely on a CFL-type stability condition which involves \(|\varphi |_{C^{0,1}}^{-1}\) , and this condition will blow up if \(\varphi \)  is not Lipschitz continuous. In the pure Hölder case, we would then need a further approximation of the nonlinearity. We refer the reader to [1]}, [2]} for details.

